Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lduallvec | Structured version Visualization version Unicode version |
Description: The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 18623; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.) |
Ref | Expression |
---|---|
lduallvec.d | LDual |
lduallvec.w |
Ref | Expression |
---|---|
lduallvec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lduallvec.d | . . 3 LDual | |
2 | lduallvec.w | . . . 4 | |
3 | lveclmod 19106 | . . . 4 | |
4 | 2, 3 | syl 17 | . . 3 |
5 | 1, 4 | lduallmod 34440 | . 2 |
6 | eqid 2622 | . . . 4 Scalar Scalar | |
7 | eqid 2622 | . . . 4 opprScalar opprScalar | |
8 | eqid 2622 | . . . 4 Scalar Scalar | |
9 | 6, 7, 1, 8, 2 | ldualsca 34419 | . . 3 Scalar opprScalar |
10 | 6 | lvecdrng 19105 | . . . . 5 Scalar |
11 | 2, 10 | syl 17 | . . . 4 Scalar |
12 | 7 | opprdrng 18771 | . . . 4 Scalar opprScalar |
13 | 11, 12 | sylib 208 | . . 3 opprScalar |
14 | 9, 13 | eqeltrd 2701 | . 2 Scalar |
15 | 8 | islvec 19104 | . 2 Scalar |
16 | 5, 14, 15 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cfv 5888 Scalarcsca 15944 opprcoppr 18622 cdr 18747 clmod 18863 clvec 19102 LDualcld 34410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-drng 18749 df-lmod 18865 df-lvec 19103 df-lfl 34345 df-ldual 34411 |
This theorem is referenced by: lkreqN 34457 lkrlspeqN 34458 lcdlvec 36880 |
Copyright terms: Public domain | W3C validator |